In this formula some of the elements of Young diagrams can be equal to zero. To prove this statement we have to explain which representations appear in the tensor product of representations. Basically [1]}, \([a_1,\ldots , a_{n+1}] \in [p_1,\ldots ,p_{n}]\otimes [r]\)  if and only if
\(\left\lbrace \begin{array}{l}a_i\le p_i+r\\p_{i-1}\ge a_i \ge p_{i}\\a_1 \ge r\\\sum a_i=r+\sum p_i\end{array}\right.\) 
